Let \({\mathbf k}\) be a finite field of characteristic \(p\) and \(G=\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\). If \(\overline\rho: G\rightarrow \text{GL}_2({\mathbf k})\) is a linear representation one would like to find a noetherian local ring \(A\) of characteristic 0 having \({\mathbf k}\) as a quotient and a continuous representation \(\rho:G\rightarrow \text{GL}_2(A)\) reducing to \(\overline\rho\). If \(S\) is some finite set of primes of \({\mathbb Q}\) and if \(\overline\rho\) factors through the Galois group \(G_S\) of a maximal algebraic extension of \({\mathbb Q}\) unramified outside \(S\) then such lifts do exist and there even is a universal deformation ring. This is a result of \textit{B.\ Mazur} [cf.\ Galois groups over \({\mathbb Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)]. An obvious refinement one would like to find for Mazur's result is to ask for a minimal ring \(A\) lifting \(\overline\rho\). In this direction the author proves the following theorem: Assume that \(p\geq 7\) and \(\overline{\rho}(G_S)\) contains SL\(_2({\mathbf k})\), where \(S\) is suitably chosen, and that several technical conditions are satisfied. Then, if \(r:=\dim_{\mathbf k}H^2(G_S,Ad^0\overline{\rho}),\) there exists a set \(Q\) of \(r\) primes outside \(S\) such that on \(G_{S\cup Q}\) one has a lift of \(\overline{\rho}\) to GL\(_2(W({\mathbf k})),\) where \(W({\mathbf k})\) is the Witt-ring of \({\mathbf k}.\) For the proof one first shows that the group \(H^2(G_v,Ad^0\overline{\rho})\) is trivial for \(v\in S.\) This implies that the group \(H^2(G_S,Ad^0\overline{\rho})\) which encodes the obstruction to liftability is isomorphic to \(\text{ Ш}^2_S(Ad^0\overline{\rho})\). By a duality result of Poitou and Tate, this is dual to \(\text{ Ш}^1_S((Ad^0\overline{\rho})^*).\) This \(r\)-dimensional vector space is viewed as an analog of the dual Selmer group from Wiles' proof of the (semistable) Shimura-Taniyama conjecture. Following this, one carefully chooses the set \(Q\) in order to kill the obstruction.